STOCHASTICITY FOR RANDOM GRAPHSJan Reimann, Penn State

June 18, 2015

RANDOM GRAPHS

Fix . is a simple, undirected graph with vertices where each edge is present (indepedently) withprobability .

A natural "limit object" for is , a countable -random graph.

This is known as the Erdös-Renyi model.

0 < p < 1 �(n, p) n

p

n → 7 �(ℕ, p)p

CONUNDRUMS OF RANDOM GRAPHS

For any , two graphs and are almost surely equivalent.

There exists a computable graph on such that forevery , almost surely is isomorphic to .      [Rado]

0 < p < q < 1 �(ℕ, p) �(ℕ, q)

ℕp �(ℕ, p)

CONSEQUENCESWhat does this imply for algorithmic randomness?

We can fix a probability distribution and developrandomness for labeled graphs and try to keep it "asinvariant as possible".

Since there is a recursive copy, no approach with evenmodest computational power will include all copies ofthe random graph.

Will the randomness be "in the isomorphism"?     [Fouché]

CONSEQUENCES

We can accept the fact that a random graph is so highlysymmetric (the automorphism group is extremely rich) thatwe have a recursive copy.

The situation then seems similar to normal numbers.

They satisfy many randomness properties (particularlyfrom a dynamical point of view).

This suggests to look at random graphs from astochasticity point of view (but what is a normal graph?).

CONSEQUENCESAs we will see, both aspects are closely related.

Does algorithmic randomness (in the"classical" sense) have anything significant to

add to the picture?

RANDOM GRAPHS AS HOMOGENEOUS STRUCTURES

The reason for the rich symmetry of the random graph canbe seen in its homogeneity.

A countable (relational) structure is homogeneous ifevery isomorphism between finite substructures of extends to an automorphism of .

The Rado graph is homogeneous by virtue of the I-property:

For any there exists

for all , .

,… , , ,… ,x1 xn y1 ym zz ∼ , z ≁xi yj

1 ≤ i ≤ n 1 ≤ j ≤ m

HOMOGENEOUS STRUCTURES

Fraissé: Any homogeneous structure arises as aamalgamation process of finite structures over the samelanguage (Fraissé limits).

Examples:

,the Rado (random) graphthe universal -free graphs, (Henson)

Homogeneous structures (over finite languages) are -categorical, i.e. their theory has only one model up toisomorphism.

(ℚ, <)

Kn n ~ 3

ℵ0

RANDOMNIZED CONSTRUCTIONS

Many homogeneous structures can obtained (almostsurely) by adding new points according to a randomizedprocess.

: add the -th point between (or at the ends) ofany existing point with uniform probability .

Rado graph: add the -th vertex and connect to everyprevious vertex with probability (uniformly andindependently).

Vershik: Urysohn space, Droste and Kuske: universalposet

Henson graph: ???

(ℚ, <) n1/n

np

CONSTRUCTIONS "FROM BELOW"

A naive approach to "randomize" the construction of theHenson graph would be as follows:

In the -th step of the construction, pick a one-vertexextension uniformly among all possible extensions thatpreserve -freeness.

However: Erdös, Kleitman, and Rothschild showed that (as goes to ) almost all graphs missing a are bipartite.

The Henson graph(s), in contrast, has to contain everyfinite -free graph as an induced subgraph, inparticular, and hence cannot be bipartite.

n

Kn

n 7 Kn

KnC5

CONSTRUCTIONS "FROM ABOVE"

Petrov and Vershik (2010) showed how to constructuniversal -free graphs probabilistically by sampling themfrom a continuous graph.

These continuous graphs, known as graphons, have beenstudied extensively over the past decade.

See, for example the recent book by Lovasz, Largenetworks and graph limits (2012).

Kn

GRAPHONS

One basic motivation behind graphons is to capture theasymtotic behavior of growing sequences of dense graphs,e.g. with respect to subgraph densities.

While the Rado graph can be seen as the limit object of asequence of finite random graphs, it does notdistinguish between the distributions with which the edgesare produced.

For any , "converges" almost surely to(an isomorphic copy of) the Rado graph.

However, , will exhibit very differentsubgraph densities than

( )Gn

0 < p < 1 �(n, p)

�p1 p2 �(n, )p1�(n, )p2

CONVERGENCE

Let be a graph sequence with .

We say converges if

( )Gn |V( )| → 7Gn

( )Gn

for every finite graph , the relative number of embeddings of into

converges.

F(F, )ti Gn F Gn

QUASIRANDOM GRAPHS

A sequence of graphs , is quasirandom if forevery graph on vertices,

That means every fixed finite graph occurs with the "right"frequency.

Hence quasirandom sequences converge in the abovesense.

( )Gn | | = nGnF k

(F, ) a ��������������.ti Gn 2+( )k

2

QUASIRANDOM GRAPHS

Quasirandom graph sequences form a natural analog tonormal sequences.

However, the additonal structure of graphs makes themmore robust. Chung, Graham and Wilson (1989) showedthat it suffices to satisfy the asymptotic frequencycondition for (one edge) and (squares) only.

One can take quasirandom graphs as a basis for "classical"stochasticity for graphs.

How robust are they under various kinds of selection rules?

This is an ongoing project of Penn State graduatestudent Jake Pardo.

K2 C4

GRAPHONS

measurable, and for all ,

and .

Think: is the probability there is an edge between and .

Subgraph densities:

edges:

triangles:

this can be generalized to define .

W : [0, 1 → [0, 1]]2 x, y

W(x, x) = 0 W(x, y) = W(y, x)

W(x, y) xy

D W(x, y) dx dyD W(x, y)W(y, z)W(z, x) dx dy dz

(F, W)ti

GRAPHONS AND GRAPH LIMITSBasic idea: "pixel pictures"

from Lovasz (2012), Large networks and graph limits

GRAPHONS AND GRAPH LIMITSConvergence of pixel pictures

from Lovasz (2012), Large networks and graph limits

GRAPHONS AND GRAPH LIMITSConvergence of pixel pictures

from Lovasz (2012), Large networks and graph limits

THE LIMIT GRAPHONTHM: For every convergent graph sequence there

exists (up to weak isomorphim) exactly one graphon suchthat for all finite :

( )GnW

F

.(F, ) ⟶ (F, W)ti Gn ti

THE LIMIT GRAPHONExample: Uniform attachment graphs

from Lovasz (2012), Large networks and graph limits

from Lovasz (2012), Large networks and graph limits

A COMPATIBLE METRICEdit distance: . Cut distance: , where is thecut norm

can be extended to graphs of different order... ... and to graphons:

(F, G) => +d1 AF AG>1

(F, G) => +d◻ AF AG>◻ >. >◻

>A = .>◻1n2 max

S,T�[n]<< ∑i!S,j!T

Aij<<

d◻

>W = W(x, y) dx dy.>◻ supS,T�[0,1] ∫S×T

   

A COMPATIBLE METRIC

A sequence converges iff it is a Cauchy sequence withrespect to . 

    iff    

( )Gnd◻

→ WGn ( , W) → 0d◻ Gn

SAMPLING FROM GRAPHONS

We can obtain a finite graph from by(independently) sampling points from andfilling edges according to probabilities .

almost surely, we get a sequence with .

If we sample -many points from , wealmost surely get the random graph.

�(n, W) Wn , … ,x1 xn W

W( , )xi xj

�(n, W) → W

ω W(x, y) z 1/2

THE PETROV-VERSHIK GRAPHON

Petrov and Vershik (2010) constructed, for each , agraphon such that we almost surely sample a Hensongraph for .

The graphons are (necessarily) {0,1}-valued.

Such graphons are called random-free.

The constructions resembles a finite extensionconstruction with simple geometric forms, where eachstep satisfies a new type requiring attention.

The method can also be used to construct random-freegraphons from which we sample the Rado graph.

n ~ 3Wn

INVARIANT MEASURES

The Petrov-Vershik graphon also yields a measure on theset of countable infinite graphs concentrating on the set ofuniversal, homogeneous -free graphs.

This measure will be invariant under the "logic action", thenatural action of on the space of countable (relational)structures with universe .

This method was generalized by Ackerman, Freer, and Patel(2014) to other homogeneous structures.

It can be used to define algorithmic randomness for suchstructures (as suggested by Nies and Fouché).

Kn

S7ℕ

UNIVERSAL GRAPHONSA random-free graphon is countably universal if for everyset of distinct points from , , ,the intersection

has non-empty interior. Here is the neighborhood of .

For countably -free universal graphs, we require this tohold only for such tuples where the induced subgraph bythe has no induced -subgraph,

additionally, require that there are no n-tuples in Xwhich induce a .

[0, 1] , , … ,x1 x2 xn , … ,y1 ym

B⋂i,j

Exi ECyj

= {y: W(x, y) = 1}Ex x

Kn

xi Kn+1

Kn

THE TOPOLOGY OF GRAPHONS

Neighborhood distance:

and mod out by .

Example: is a singleton space.

THM: (Freer & R.) (informal) If is a random-free universalgraphon obtained via a "tame" extension method, then is notcompact in the topology.

(x, y) = > W(x, . ) + W(y, . ) = ∫ |W(x, z) + W(y, z)|rW >1

(x, y) = 0rW

W(x, y) z p

WW

rW

"TAME" EXTENSIONSDEF: A random-free graphon has continuous realizationof extensions if there exists a function

that is continuous a.e. such that for all ,

Here is the neighborhood of .  The Petrov-Vershik graphons have uniformly continuousrealization of extensions.

W

f : ( , … , ), ( , … , ) ↦ (l, r)x1 xn y1 ym,x ⃗ y ⃗

[l, r] � B .⋂i,j

Exi ECyj

= {y: W(x, y) = 1}Ex x

NON-COMPACTNESS  

THM: If a countably ( -free) universalgraphon has uniformly continuous realization

of extensions, then it is not compact in the -topology.

Kn

rW

FEATURES OF THE PROOF

Building a "Cantor sequence" in .

Apply the Szemeredi regularity lemma to pass to asequence of stepfunctions that approximate the graphonuniformly.

Use universality to find the next splitting.

Uniform continuity guarantees that the Szemeredi"squares" are filled with the right measure.

W

REGULARITY LEMMA

For every there is an such that everygraph with at least vertices has an equitablepartition of V into pieces (1/ε ≤ k ≤ S(ε)) such that for allbut pairs of indices , the bipartite graph is

-regular.

For every graphon and there is stepfunction with steps such that

ϵ > 0 S(ϵ) ! ℕG S(ϵ)

kϵk2 i, j G[ , ]Vi Vj

ϵW k ~ 1 U

k

(W, U) < > Wd◻2

log k‾ ‾‾‾‾3>2

COMPLEXITY OF UNIVERSAL GRAPHONSconstruction: fully

randomtamedeterministic

generaldeterministic

complexity ofgraphon

low(singleton)

high (non-compact,infiniteMinkowskidimension)

           ?

Also: Is there a robust notion of a stochasticgraphon?